

The claim is stated for a fixed horizon \(T\)  in [1]}. However, we can define the surrogate random variable
\(X^{\prime }_i \triangleq X_i\cdot \left(1-\mathbb {I}\left\lbrace \exists 0\le j<i:\, \sum _{s=1}^jX_s > \eta \sum _{s=1}^j\mathbb {E}[X_s^2\,|\,\mathcal {F}_s]+\eta ^{-1}\ln (\delta ^{-1})\right\rbrace \right).\) 
